An adaptive moving grid finite difference method

The grid generation is very crucial for the accuracy of the numerical solution of PDEs, especially for problems with very rapid variations or sharp layers, such as shock waves, wing leading and trailing edges, regions of separation, and boundary layers. The adaptive grid generation is an iterative approach to accommodate these complex structures. In this paper, we introduce a deformation based adaptive grid generation method, in which a differentiable and invertible transformation from computational domain to physical domain is constructed such that the cell volume (Jacobian determinant) of the new grid is equal to a prescribed monitor function. A vector field is obtained by solving the div-curl system and can be used to move the grids to the desired locations. By computing the inverse of Jacobian, any deformed grids can also be transformed back to the uniform grid. Several numerical results in two dimensions are presented. Some applications in image registration are discussed.

A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution

<p style='text-indent:20px;'>In this work a novel mesh adaptation technique is proposed to approximate discontinuous or boundary layer solution of partial differential equations. We introduce new estimator and monitor function to detect solution region containing discontinuity and layered region. Subsequently, this information is utilized along with equi-distribution principle to adapt the mesh locally. Numerical tests for numerous scalar problems are presented. These results clearly demonstrate the robustness of this method and non-oscillatory nature of the computed solutions.</p>

Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling

In the present article, we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with an appropriate predictor–corrector method to achieve higher resolutions. The underlying finite volume scheme is conservative, and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed; thus, unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according to the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.

Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling

In the present article, we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with an appropriate predictor–corrector method to achieve higher resolutions. The underlying finite volume scheme is conservative, and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed; thus, unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according to the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.

Numerical treatment for the solution of singularly perturbed pseudo-parabolic problem on an equidistributed grid

The initial-boundary value problem for a pseudo-parabolic equation exhibiting initial layer is considered. For solving this problem numerically independence of the perturbation parameter, we propose a difference scheme which consists of the implicit-Euler method for the time derivative and a central difference method for the spatial derivative on uniform mesh. The time domain is discretized with a nonuniform grid generated by equidistributing a positive monitor function. The performance of the numerical scheme is tested which confirms the expected behavior of the method. The existing method is compared with other methods available in the recent literature.

An adaptive moving mesh method for a time-fractional Black–Scholes equation

In this paper we study the numerical method for a time-fractional Black–Scholes equation, which is used for option pricing. The solution of the fractional-order differential equation may be singular near certain domain boundaries, which leads to numerical difficulty. In order to capture the singular phenomena, a numerical method based on an adaptive moving mesh is developed. A finite difference method is used to discretize the time-fractional Black–Scholes equation and error analysis for the discretization scheme is derived. Then, an adaptive moving mesh based on an a priori error analysis is established by equidistributing monitor function. Numerical experiments support these theoretical results.

Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling

In the present article, we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with an appropriate predictor–corrector method to achieve higher resolutions. The underlying finite volume scheme is conservative, and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed; thus, unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according to the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.

Application End of Line Resistor (EOLR) and Voltage Divider for Passive Infrared Motion Sensor

End of Line Resistor (EOLR) is a resistor that has a certain value to protect the circuit at the end of a loop or zone. The End of Line Resistors is used in security systems to enable the control panel to monitor wiring of open and closed. In this research using an Arduino microcontroller and Passive Infrared (PIR) Motion Sensor that connected with the systematic circuit to supervisory a loop or zone. Systematic circuit use combination of the resistor as the end of line resistor and voltage divider method. Analog input on Arduino uses to detect different states of sensors. The 1K resistor in series with the sensor line provides some protection for the input pin against the unexpected current flow. The 4K7 pull-up resistor, when combined with the resistance provided by the sensor, acts as a voltage divider, exposing the analog input to a different voltage depending on the state of the sensor. The result of measurement viewed by serial monitor function on the personal computer. PIR Sensor connected with an analog pin on microcontroller running well and there is no significant voltage value changing.

Voltage Protection and Harmonics Cancellation in Low Voltage Distribution Network

Nowadays low voltage distribution network is considered as worldwide future generation distribution network. But the major concern is harmonics generation and steps taken to cancel those harmonics. In our proposed work, low voltage distribution network is designed with low voltage and harmonics are cancelled in our method. The combination of current control unit and voltage control unit will give extra reliable power solution to increase the required capacity of low voltage grids. The high voltage protection gears are used in worst environment for low voltage and low current distribution network test is preferable to assess a variety of operation uniqueness. Therefore, it has few restrictions in implementation of economic in addition to process methodologies. In our work a 48V direct current base up-scale low voltage distribution network test is urbanized to allow the copy and surveillance of a variety of phenomenon of direct current distribution networks. The proposed system provide stretchy pattern ability by introduce S-connectors and T-connectors module that will be proscribed distantly, and near real time monitor function through by means of a data acquisition system associated toward the nodes. Each connector be able to calculate Power, Voltage and current with up to 250 kHz frequency. To calculate power quality and to understand the performance of the distribution network, frequency analysis is required along with collected data.